Question

Question 6 A3 A gas moves according to the (non-dimensional) continuity and mnomentum 12 marks equations - ρυ) 0, at + v- at where is the density(v)is the velocity and thé stress is σ-cos (πρ ) . Verify that v, = 0 and p, = 1/2 is an equilibrium of the gas flow. Introduce small variations of the velocity about v, and density about p., linearise the PDES and derive the wave equation governing the evolution of the density variations. What is their speed of propagation? (For your reference: sin(a+ b) sin a cos b+ cos a sin b).

Answer 1

Sol: JPu)O P +Vav where p is the density vs the velocily and the stress is 6=COs (P) pVis an eauiibrium of the gas Phw. V-vav KP with cos (TS) we Put vao b S 2 6 (05 (7)2) -0 and so e (cauilibrlum of gas Plow) as 0 Now vavt V and e + - a'52 dt 2 dx

dxEvsV2ldn v-7 cos () as t v (2) JV4ds in T'a 2-T COS Ix +T1 + In higher order terms There arc To be negected 2.2 here V? JEn (os (75 Take time deriation of the density cauation of uscd second eauation for subhsbitution